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Scholarly Works (7 results)

  • Thesis
  • Peer Reviewed
UC Berkeley Electronic Theses and Dissertations (2019)

Multilinear integral inequalities, such as H¨older’s inequality and Young’s convolution inequality, play a large role in analysis. In [6] and [5], Bennett, Carbery, Christ, and Tao provide a classification of such inequalities of the form

 → R. Rd

n j=1

fj(Lj(x))dx ≤ C

n j=1

||fj||pj

for constant C > 0, exponents pj ∈ [1,∞], and surjective linear maps Lj : Rd → Rdj with fj : Rdj

, In Chapter 2, we discuss a generalization of the above related to work by Ivanisvili

and Volberg [23]. Specifically, we provide a classification of functions B : Rn → [0,∞) such that

 B(f1(L1(x)), ..., fn(Ln(x))dx ≤ CB Rd  f1, ...,

  fn

.

In some cases, it will be shown that maximizers of the above inequality exist. Tuples of Gaussians are not always maximizers, differing from the usual multilinear theory. Chapter 3 focuses its attention on the trilinear form for twisted convolution:

 f(x)g(y)h(x + y)eiσ(x,y)dxdy. Rd×Rd While existence of maximizers can shed light on the structure of an operator,

sometimes it is useful to establish more refined information. For twisted convolution, we show a quantitative version of the statement that if a triple of functions nearly maximizes the form, then it must be close to a maximizing triple. Such a statement2

may be referred to as a sharpened inequality. Here, the proof of a sharpened inequality is complicated by the fact that no maximizers exist for twisted convolution; however, one may vary the amount of oscillation and compare to the case in which there is zero oscillation. In Chapter 4, we establish a sharpened version of the following inequality due to

Baernstein and Taylor [3]: 

Sd×Sd f(x)g(y)h(x · y)dσ(x)dσ(y) ≤  f∗(x)g∗(y)h(x · y)dσ(x)dσ(y) Sd×Sd

where f, g, h are restricted to the class of indicator functions and h is monotonic on [−1, 1]. In the above, f∗ refers to the symmetric decreasing rearrangement of f, and likewise for g and g∗.

    Cover page: Multilinear and Sharpened Inequalities
    • Article
    Proceedings of the Annual Meeting of the Cognitive Science Society, Volume 40 (2018)
    We provide a new computational model of working memory inthe complex span task implemented in the ARCADIA cogni-tive framework. While there exist implementations of workingmemory successful enough to account for many of the bench-mark findings in the working memory literature, we demon-strate that further progress requires the integration of thesemodels with a rich conception of attention. ARCADIA pro-vides this intersection, allowing for precise control of the focusof attention on a time scale fine enough to begin to disentan-gle the overlapping effects of interference, temporal decay, andattentional refreshing.
      Cover page: Time-Based Resource Sharing in ARCADIA
      • Article
      • Peer Reviewed
      Proceedings of the Annual Meeting of the Cognitive Science Society, Volume 44 (2022)
      There are conflicting theories about how people reason through cause and effect. A key distinction between two prominent accounts pertains to whether, in judging an event’s causal relevance, people preferentially consider what actually happened (as predicted by process theories) or whether they also consider what could have happened under different conditions (as predicted by counterfactual theories). Toward adjudicating between these theories, the current work used eye tracking and Gaussian Process modeling to investigate how people form causal judgments retrospectively and in the absence of ongoing visual input. Participants played a virtual ball-shooting game: after choosing to move left or right, they encoded a video of the actual outcome and then were prompted to mentally simulate either (a) what actually happened, (b) what could have happened, or (c) what caused the outcome to happen while looking at a blank screen. During causal judgment, we found evidence that participants visually mentally simulated counterfactual possibilities: they moved their eyes in similar patterns as when they imagined a counterfactual alternative. Altogether, these results favor counterfactual theories of causal reasoning, demonstrate how visual mental simulation can support this reasoning, and provide a novel methodological approach for using eye movements to investigate causal reasoning and counterfactual thinking more broadly.
        Cover page: Eye-tracking mental simulation during retrospective causal reasoningCreative Commons 'BY' version 4.0 license
        • Article
        • Peer Reviewed
        Proceedings of the Annual Meeting of the Cognitive Science Society, Volume 44 (2022)
        The human capacity for causal judgment has long been thought to depend on an ability to consider counterfactual alternatives: the lightning strike caused the forest fire because had it not struck, the forest fire would not have ensued. To accommodate psychological effects on causal judgment, a range of recent accounts of causal judgment have proposed that people probabilistically sample counterfactual alternatives from which they compute a graded index of causal strength. While such models have had success in describing the influence of probability on causal judgments, among other effects, we show that these models make further untested predictions: probability should also influence people's metacognitive confidence in their causal judgments. In a large (N=3020) sample of participants in a causal judgment task, we found evidence that normality indeed influences people's confidence in their causal judgments and that these influences were predicted by a counterfactual sampling model. We take this result as supporting evidence for existing Bayesian accounts of causal judgment.
          Cover page: Measuring and Modeling Confidence in Human Causal JudgmentCreative Commons 'BY' version 4.0 license
          • Article
          Proceedings of the Annual Meeting of the Cognitive Science Society, Volume 40 (2018)
          Herein we describe CRAMM, a framework for Causal Reason-ing via Attention and Mental Models. CRAMM develops andextends assumptions made by a previously developed coun-terfactual simulation model of human causal judgment. Weimplement CRAMM computationally and demonstrate how itrobustly captures human causal judgments about simple two-object interactions at the level of underlying cognitive and per-ceptual processes, including data on eye-movements that serveas direct evidence for the role of counterfactuals in causal judg-ment.
            Cover page: An Attention-Driven Computational Model of Human Causal Reasoning
            • Article
            • Peer Reviewed
            Proceedings of the Annual Meeting of the Cognitive Science Society, Volume 46 (2024)
            Imagine that a river becomes polluted if two plants generate too much waste. One might be more inclined to say that a plant caused the river to become polluted when it produced more waste than expected. While similar normality effects on causal judgments have been observed in cases with binary variables, little work has focused on cases with continuous variables. To test whether the statistical normality of continuous variables influences causal judgments, we had participants learn statistical norms over repeated iterations of a vignette and make a causal judgment about an instance of that vignette. Following Icard et al. (2017), we manipulated the causal structure and the normality of each cause. By testing whether normality effects on causal judgment generalize to cases with continuous variables, our results help determine whether these effects are central to human cognition, or simply apply to a subset of cases studied thus far.
              Cover page: Causation on a continuum: normality effects on causal judgmentsCreative Commons 'BY' version 4.0 license
              • Article
              • Peer Reviewed
              Proceedings of the Annual Meeting of the Cognitive Science Society, Volume 41 (2019)
              Previous studies have shown that category learning affects subsequent recognition memory. However, questions remain as to how category learning affects discriminability during recognition. In this three-stage study, we employed sets of simulated flowers with category- and non-category-inclusion features appearing with equal probabilities. In the learning stage, participants were asked to categorize flowers by identifying the category-inclusion feature. Next, in the studying stage, participants memorized a new set of flowers, a third of which belonged to the learned category. Finally, in the testing stage, participants received a recognition test with old and new flowers, some from the learned category, some from a not-learned category, some from both categories, and some from neither category. We applied hierarchical Bayesian signal detection theory models to recognition performance and found that prior category learning affected both discriminability as well as criterion bias. That is, people that learned the category well, exhibited improved discriminability and a shifted bias toward flowers from the learned relative to the not learned category.
                Cover page: The Effect for Category Learning on Recognition Memory: A Signal Detection Theory Analysis
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